We study a system of spherical non-adsorbing particles immersed in a polydisperse polymer fluid. We derive an analytic expression for the many-body depletion interactions between the colloidal particles in the limit of very long chains. We argue that this expression is essentially exact for long chains and justify this using explicit simulations. In this way we are able to elucidate the profound effect of many-body interactions on the particle thermodynamics. We show that using truncated 2-body depletion interactions leads to strong particle segregation, while the complete many-body description predicts that the total depletion force becomes weak so that the system approximates one which interacts via a so-called Kac potential. This implies that the depletion interactions can be described using mean-field theory. We show that many-body effects cause a significant contraction of the 2-phase region of the particle dispersion. We also investigate the system approaching the (tricritical) θ point, which terminates the line of first-order critical points of the polymer dispersion in a poor solvent and show that many-body effects suppress particle phase transitions.